Now suppose \(X_0,X_1,X_2,\ldots \)  is a (time-homogeneous) Markov chain in \(E\)  with transition matrix \(\mathbf {P}\) . In particular, \(P(x,y)={\rm P}(X_{t+1}=y\mid X_t=x)\)  for all \(x,y\in E\)  and \(t=0,1,2,\ldots \) . Under what conditions on \(\mathbf {P}\)  is \([X_0],[X_1],[X_2],\ldots \)  a Markov chain in the “reduced” state space \(\bar{E}\) ? A sufficient condition, apparently due to Kemeny and Snell [1]}, is that \(\mathbf {P}\)  be lumpable with respect to \(\sim \) . By definition, this means that, for all \(x,x^{\prime },y\in E\) ,
\(x\sim x^{\prime }\quad \text{implies}\quad \sum _{y^{\prime }\in [y]}P(x,y^{\prime })=\sum _{y^{\prime }\in [y]}P(x^{\prime },y^{\prime }).\) 
